Asymptotic stability of sharp fronts. I One bound state implies stability

We study the asymptotic stability of the traveling front solutions to nonlinear diffusivedispersive equations of Burgers type. Our main example is the Korteweg-de Vries–Burgers (KdVB) equation although the result holds much more generally. Exploiting the modulation of the translation parameter of the front solution and making an energy estimate, we establish our stability criterion that a certain Schrödinger equation in one dimension has exactly one bound state. Counting the number of bound states of the Schrödinger equation, we find a sufficient condition for stability in terms of a Bargmann’s integral, which can be interpreted as the distance of the monotonization of the front and the corresponding ideal shock. For the KdVB equation, we analytically verify that our sufficient condition is met in an open interval of the relative dispersion parameter that includes all monotone fronts. Numerical experiments suggest stability for a much larger interval.